On the generalized Lebedev index transform

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel $|K_{(i\tau+\alpha)/2}(x)|^2, \alpha \ge 0,\ x >0, \ \tau \in \mathbb{R},$ where $K_\mu(z)$ is the Macdonald function and $i$ is the imaginary unit. Mapping properties such as the boundedness, compactness, invertibility are investigated for these operators and their adjoints in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. As an interesting application, a solution of the initial value problem for the second order differential difference equation, involving the Laplacian, is obtained

Lebedev's type index transforms with the modified Bessel functions

New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the modulus of Macdonald's function, recently considered by the author.Comment: arXiv admin note: text overlap with arXiv:1509.0872